Duchenne muscular dystrophy (DMD) is an X-linked dystrophin-minus muscle-wasting disease. Ion homeostasis in skeletal muscle fibers underperforms as DMD progresses. But though DMD renders these excitable cells intolerant of exertion, sodium overloaded, depolarized, and spontaneously contractile, they can survive for several decades. We show computationally that underpinning this longevity is a strikingly frugal, robust Pump-Leak/Donnan (P-L/D) ion homeostatic process. Unlike neurons, which operate with a costly “Pump-Leak–dominated” ion homeostatic steady state, skeletal muscle fibers operate with a low-cost “Donnan-dominated” ion homeostatic steady state that combines a large chloride permeability with an exceptionally small sodium permeability. Simultaneously, this combination keeps fiber excitability low and minimizes pump expenditures. As mechanically active, long-lived multinucleate cells, skeletal muscle fibers have evolved to handle overexertion, sarcolemmal tears, ischemic bouts, etc.; the frugality of their Donnan dominated steady state lets them maintain the outsized pump reserves that make them resilient during these inevitable transient emergencies. Here, P-L/D model variants challenged with DMD-type insult/injury (low pump-strength, overstimulation, leaky Nav and cation channels) show how chronic “nonosmotic” sodium overload (observed in DMD patients) develops. Profoundly severe DMD ion homeostatic insult/injury causes spontaneous firing (and, consequently, unwanted excitation–contraction coupling) that elicits cytotoxic swelling. Therefore, boosting operational pump-strength and/or diminishing sodium and cation channel leaks should help extend DMD fiber longevity.

Overview

Skeletal muscle (SM) fibers (SMFs) constitute ∼40% of human body mass (Janssen et al., 2000). The sarcolemma of these mechanical, long-lived, multinucleate cells is fortified by dystrophin, a filamentous entropic–spring-like membrane skeleton protein (Campbell and Kahl, 1989; Ibraghimov-Beskrovnaya et al., 1992; Khairallah et al., 2012; Constantin, 2014; Allen et al., 2016; Le et al., 2018). Dystrophin protects against sarcolemmal tearing and bleb damage; with bleb damage, membrane skeleton/bilayer adhesions detach, resulting in fragility and the decay of cell-mediated bilayer organization (Methfessel et al., 1986; Sheetz et al., 2006; Lundbaek et al., 2010; García-Pelagio et al., 2011; Morris, 2012; Ameziane-Le Hir et al., 2014; Dos Santos Morais et al., 2018; Burden et al., 2018). Via linkages within the massive trans-sarcolemma dystrophin–glycoprotein complex, dystrophin also facilitates physiological signaling (Dowling et al., 2021). Healthy SMF capillary beds require dystrophin-expressing endothelial and smooth muscle cells (Verma et al., 2019; Podkalicka et al., 2019). Dystrophin-expressing thymus-derived cells contribute to inflammatory repair of injured muscle (e.g., Farini et al., 2021). Sporadically, damaged SMFs regenerate by fusion with their stem (satellite) cells, a process reliant on satellite cell dystrophin (Dumont et al., 2015; Filippelli and Chang, 2021).

Individuals with the X-linked muscle-wasting disease Duchenne muscular dystrophy (DMD; Fig. 1 A) have no functional dystrophin. Non-ambulatory by their teens, DMD patients can nevertheless survive until SMFs of the respiratory system become nonviable, i.e., for 3 to 4 decades (Landfeldt et al., 2020). With its gene discovered (Hoffman et al., 1987), dystrophin was localized at the sarcolemma (Zubrzycka-Gaarn et al., 1988), and the dystrophin-minus mouse (mdx) was established as a dystrophin-minus mouse model for DMD (Ryder-Cook et al., 1988). DMD remains incurable (Bishop et al., 2018; Datta and Ghosh, 2020; Duan et al., 2021). Though chronic Na+ overload of dystrophic muscles (first documented in 1955) is now noninvasively detectable via 23Na–magnetic resonance imaging (MRI) in young patients (Table 1, items 1, 5, 6, 9, and 10), how SMF ion homeostasis falters in DMD, and if/how this relates to fiber loss, remain unclear.

Ion homeostasis is the autonomous Pump-Leak/Donnan (P-L/D) feedback process (Fig. 2) by which cells, after ionic perturbations, reestablish their “set point” (i.e., steady-state values for membrane potential [Vm] + volume cell model parameter [Volcell] + [ions]i). Fig. 2 A outlines the “P-L and the D-mediated” feedbacks of ion homeostasis. Different cell types’ particular set points are established by evolutionary history. Though concepts depicted in Fig. 2 B are not new, new terms are introduced there to the ion homeostasis lexicon: P-L/D systems, minimal (versus nonminimal) P-L/D steady states, Pump-Leak–dominated steady states, and Donnan-dominated steady states.

The charge difference (CD) approach for P-L/D ion homeostasis modeling is rigorous and powerful (Fraser and Huang, 2004, 2007; Fraser et al., 2011; Cha and Noma, 2012; Hübel et al., 2014; Dijkstra et al., 2016; Kay, 2017; Dmitriev et al., 2019) but has not yet been used to address DMD-afflicted fibers, which, in spite of an encyclopedic list of deficits (e.g., Murphy et al., 2019; Dowling et al., 2021), can nevertheless survive decades. Here, via CD-modeling, we therefore ask not only what is wrong with DMD ion homeostasis but also what is right. Globally, we conclude, what is wrong relates to Pump-Leak malfunctions, while what is right relates to SMFs’ ion homeostatic strategy, which, unlike that of neurons, is built around an ultra–energy-efficient and thence physiologically robust Donnan dominated (i.e., based on [big PCl][small INaleak]) steady-state. Cytoplasmic Donnan effectors are a source of electro-osmotic free energy; as membrane-impermeant ions, they passively influence the transmembrane distribution of membrane-permeant ions and of H2O (Fig. 2, A and B). Neurons are high-input–impedance, electrically agile, high-excitability cells with [small PCl][big INaleak] steady states; when pumping falters, neurons’ [big INaleak] and resultant small pump reserves quickly eventuate in treacherous Donnan-effector–mediated swelling (= lethal Na+ + Cl + H2O influxes). SMFs, too, are excitable, but they spend most of their time at their hyperpolarized steady states. SMF resilience against profound physiological insult starts from the collaboration of [big PCl] with Donnan effectors, a pairing that exploits the no-added-cost free energy of Donnan effectors to keep SMFs difficult to excite. Meanwhile, [small INaleak] ensures hyperpolarized resting potential (Vrest) values, slow rundowns when pump strength falters, and the large resilience-conferring pump reserves that ensure SMFs can handle intermittent peak demands.

Operational “pump strength” defined

Ion homeostatic pump strength here is an operational term, given as percentage of normal (100%) maximal pump strength. Pump-strength units are attomol/s (amol/s; 10−18 mol/s) ATP consumed or, equivalently, pA of INaKpump (see Materials and methods). Facilitating intermodel comparisons, for all P-L/D models here, maximal (100%) pump strength = 566 amol/s of ATP consumption (i.e., 54.5 pA of INaKpump for ImaxNaKpump) and pump kinetic parameters are identical. A P-L/D system’s operational pump strength would, then, depend, in vivo, on (1) the quantity of pump proteins (an expression issue), (2) the functionality of membrane-resident pumps (a “machinery” issue), and (3) the supply issue of ATP availability (i.e., [oxygen-glucose→→→ATP]). Thus (hormonal) up-regulation would give pump strengths >100% whereas (e.g.) tourniquet application, chronic or episodic ischemia, malfunctioning pumps, ouabain, vascular deterioration, etc., would give pump strengths <100%. As DMD progressed, cumulative defects of the dystrophin-minus condition (Fig. 1, A–C) would diminish pump strength, thus defined.

Reduced pump strength and elevated Na+ leaks in DMD

The mitochondrial damage of DMD is first a pump-strength supply issue, and second, from reactive oxygen species (ROS) and Ca2+ damage to pump-bearing sarcolemma, a machinery issue (Whitehead et al., 2010; Timpani et al., 2015; Moore et al., 2020; Dubinin et al., 2020; Ramos et al., 2020; Capitanio et al., 2020). A further supply issue is inadequate DMD vasculature (Dietz et al., 2020). Because of early reports of above-normal Na+/K+-adenosine triphosphatase (ATPase) protein levels in mdx SMFs (Anderson 1991; Dunn et al., 1995), pump expression was not considered problematic. New work shows otherwise. Kravtsova et al. (2020) find, for 3Na+/2K+ATPase isozymes in mdx respiratory (diaphragm) and postural (soleus) muscles, reduced protein (and mRNA) levels, depolarized Vrest values, and diminished ouabain depolarization. Moreover, given the sensitivity of Na+/K+-ATPases to bilayer lipids (Cornelius et al., 2015; Habeck et al., 2017; Petrov et al., 2017; Hossain and Clarke, 2019; Else, 2020), the pathologically redistributed diaphragm endplate cholesterol Kravtsova et al. (2020) observe likely further diminish pump strength as a machinery issue. Pediatric DMD patients’ dyslipidemia worsens with age, while deep remodeling of energy metabolism occurs in DMD SMFs (this includes reduced SMF ATP, a supply issue) and changes in multiple membrane-forming lipids (Anderson, 1991; White et al., 2020; Dabaj et al., 2021). These deficits are consistent with sarcolemmal damage of DMD due to mechanical and oxidative (Ca2+/ROS) stress (Petrof et al., 1993; Dudley et al., 2006; Allen et al., 2016; Murphy et al., 2019).

Acutely, during exertion, healthy SMFs locally bolster their vascular supply: a nitric oxide (NO) synthetase (NOS) linked via dystrophin to the dystrophin–glycoprotein complex activates, producing NO to vasodilate capillaries. In dystrophin-minus fibers, this feedback fails (Sander et al., 2000; Asai et al., 2007). The result is functional ischemia. This is exacerbated by DMD muscles’ degenerated vascularization (Dudley et al., 2006; Thomas, 2013; Bosco et al., 2021). Compounding these supply issues, longer-term, oxidative stress (Ca2+, ROS) from functional ischemia causes membrane damage, a machinery issue.

On the Na+-leak side, perturbed gating in bleb-damaged mdx sarcolemma could explain overactive voltage-gated sodium channel (Nav) 1.4 channels and overactive (unidentified) cation channels (Methfessel et al., 1986; Morris and Horn, 1991; Wan et al., 1999; Morris, 2012, 2018; Morris and Joos, 2016; Hirn et al., 2008; Lansman, 2015).

The capacity for sarcolemmal repair is retained

Stretch injury can leave mdx fibers depolarized and inexcitable for days but is not inherently lethal (Anderson, 1991; Call et al., 2013; Pratt et al., 2015; Baumann et al., 2020). In spite of their heightened susceptibility to rupture, microtears, and bleb damage (Menke and Jockusch, 1991, 1995; Petrof et al., 1993; Williams and Bloch, 1999; García-Pelagio et al., 2011; Hernández-Ochoa et al., 2015; Houang et al., 2018; Fig. 1 B), DMD fibers, like healthy SMFs, can repair sarcolemma tears. Ca2+-mediated exocytosis seals the tear; endocytosis then retrieves excess bilayer, though mdx-fibers’ impaired autophagy impedes subsequent reprocessing (McNeil and Steinhardt, 2003; Corrotte et al., 2013; Andrews et al., 2014; Barthélémy et al., 2018; Stoughton et al., 2018; Call and Nichenko, 2020).

Na+ overload: Though rapidly lethal in injured neurons, chronically tolerated in DMD SMFs

Na+ overload of ischemically injured central neurons rapidly elicits treacherous osmotic swelling (Hübel et al., 2014; Dreier et al., 2018). In cortical neurons (CNs), this accelerates lethally when normally cryptic Cl channels activate (Rungta et al., 2015). Dijkstra et al. (2016) explained this cerebral ischemia scenario using a neuronal P-L/D ion homeostasis model, here termed CN-CD.

23Na-proton-MRI measurements from leg muscles of preteen boys with DMD show a chronic Na+ overload that has been shown to be “nonosmotic,” i.e., not accompanied by water uptake (Table 1, items 5, 6, and 10; Weber et al., 2011, 2012; Gerhalter et al., 2019). This presents a seeming conundrum because, like their healthy counterparts, DMD-afflicted SMFs have an exceptionally large resting PCl (chloride permeability), or “[big PCl],” ∼80% of which is due to ClC-1 channels (Cozzoli et al., 2014; Pedersen et al., 2016; Jentsch and Pusch 2018). CNs’ “[small PCl]” helps minimize swelling during normal brief Na+ loads but is inadequate during sustained ischemia-induced Na+ loading. Then, for well-understood reasons, once the abnormal PCl component joins in, the neurons succumb even faster to cytotoxic [Na+ + Cl + H2O] (osmotic) influxes or “death-by-Donnan effect.” How, then, do [big PCl] DMD fibers, with their low pump strength, chronically sustain nonosmotic Na+ overloads? Though PK values differ nontrivially in neurons and SMF, the explanation does not lie there. Modeling here will show those differences to be constrained by far more critical neuron/SMF differences in PNa and PCl values.

Neurons, to support their easy-to-excite electrically agile electrophysiological lifestyle, evolved a costly Pump-Leak–dominated ([small PCl][big INaleak]) strategy for ion homeostatic steady state. SMFs, cells that spend most of their time at steady state, where they need to remain difficult-to-excite, rely on a different strategy ([big PCl][small INaleak]).

A useful P-L/D model for excitable SMFs, one that can resolve the neuron/SMF “PCl conundrum,” should explain healthy SMF ion homeostasis while clarifying how DMD-afflicted, Na+-overloaded SMFs avoid osmotic swelling. The model used here does so. It shows how SMFs’ [big PCl]/(Donnan effector) collaboration physiologically exploits free energy embodied in impermeant myoplasmic anions to deeply stabilize, for no added cost, their Pump-Leak–determined steady state. This strategy keeps SMF excitability low and extraordinarily safe from osmotic swelling, provided they maintain the [small INaleak]. Modeling likewise shows how the Donnan dominated (i.e., [big PCl][small INaleak]) SMF ion homeostatic steady state would eventually fail, as severe (late-stage) DMD damage rendered INaleak too big and global SMF pump strength too small.

Comparative CD modeling to address DMD

To probe healthy SMF volume regulation, Fraser and Huang (2004) established the biophysically rigorous CD approach to ion homeostasis modeling (Fraser and Huang, 2007; Kay, 2017; Dmitriev et al., 2019) used here. Unlike earlier approaches, it explicitly incorporates the thermodynamic feedbacks (constraints) imposed by cytoplasmic Donnan effectors (Fig. 2 A). That first Fraser and Huang (2004) P-L/D model (like models here) is 0-D, meaning that flux properties are spatially invariant. It is, however, nonexcitable. A later model addressing action potential (AP) propagation in K+-accumulating t-tubules adds spatial complexity and excitability (Fraser et al., 2011; see T-tubules in the supplemental text at the end of the PDF), but is unsuited for addressing the resilience of SMF ion homeostasis in the face of DMD.

For that purpose, we use parallel (i.e., comparative) P-L/D modeling. SM-CD is our as-basic-as-possible P-L/D model for a slice of generic excitable SMF; as per Fig. 3 A–C and Table 2 (see List of abbreviations), it parallels the extant model, CN-CD (Dijkstra et al., 2016). SM-CD has SMF-appropriate resting permeabilities and excitability, but matches CN-CD for membrane area, maximal (100%) pump strength, and steady-state volume. Parallel models make intersystem comparisons of (1) steady-state values and (2) post-perturbation trajectories biophysically meaningful. SM-CD has a minimal steady state, but CN-CD (with its neuronally important K/Cl cotransporter) has a nonminimal steady state. In all models, INaKpump (i.e., hyperpolarizing Na+ extrusion from 3Na+out/2K+in) is the direct electrophysiological consequence of ion homeostatic ATP consumption. For SM-CD, steady state is Donnan dominated; to quantitatively clarify the benefits of that steady state, an additional parallel model, weak Donnan (WD)–CD is devised: it is a counterfactual SM-CD analog with a Pump-Leak–dominated steady state.

Donnan effectors rehabilitated

It is widely and correctly understood that energetically compromised cells risk death-by-Donnan effect. It is widely underappreciated, however, that Donnan effectors are cell-physiologically indispensable (Fig. 2 A). However, because “Donnan equilibrium” (DE) equals death, Donnan has come to be a dirty word. Thus, the all-important “passive stabilization of ECl near Vrest” in SMFs is alluded to frequently (e.g., Pedersen et al., 2016), but the mechanism by which this is achieved (i.e., by exploiting the osmo-electrical free energy inherent in SMFs’ abundant cytoplasmic Donnan effectors) is not. Thus, ion homeostasis gets overlooked as a role for SMFs’ ClC-1 channels (Jentsch and Pusch, 2018). Thus, the immense importance for vertebrate evolution of SMFs’ robust efficient Donnan dominated ([big PCl][small INaleak]) steady state seems to go unnoticed.

Underappreciated, by extension, is the pivotal role of SMFs’ unidentified small-valued PNa. This oversight is explicable because Vrest is usually calculated with the GHK voltage equation (e.g., Sperelakis, 2012; and see Hille, 2001, Eq. 14.10). There, the smaller the PNa, the less consequential the tiny INaleak carried by PNa. Using P-L/D equations for SMF steady state, however, shows that small PNa carries not an inconsequentially tiny INaleak but a powerfully tiny INaleak. It is powerful because it underlies SMFs’ tiny steady-state ATP consumption.

Rehabilitating Donnan effectors’ bad reputation as we do here highlights that, for vertebrate bodies (on a body mass basis), low-cost Donnan dominated steady states (as in SMFs) are the norm. High-cost Pump-Leak dominated steady states (as in neurons) are, and always have been (on a body mass basis), the special case (Table S1).

DMD: Fiber demise and fiber survival

DMD fiber necrosis is typically ascribed to Ca2+ necrosis (Claflin and Brooks, 2008; Allen et al., 2016; Mareedu et al., 2021), but as emphasized by Burr et al. (2014), who examined mdx Ca2+ necrosis via reverse operation of Na+/Ca2+ exchangers, underlying that Ca2+ necrosis is a preexisting Na+ overload. Summary comments by Burr and Molkentin (2015) reveal that the provenance of DMD Na+ overload is poorly understood. They write that “given the known mechanical defects within the dystrophic plasma membrane…alterations in calcium and sodium levels likely stem…from excessive activation of various channels and exchangers.” While this is mechanistically vague, what is clear is their implication that too much Na+ leak would account for chronic Na+ overload. Our analysis strongly suggests, however, that for chronic Na+ overload to result in fiber loss, too little Na+ pumping (i.e., low pump strength) would be a major factor, while too much Na+ leaking would be a dangerous exacerbating factor, but not a sole cause.

Specifically, modeling here shows how low pump-strength conditions elicit unwanted Na+ entry through normal Nav channels. Additionally, DMD fibers can have pathologically leaky Nav and/or cation channels that, under low pump-strength conditions, would contribute to chronic Na+ overloads. Modeling shows, too, the vastly different thresholds for cytotoxic swelling (elicited by spontaneous firing) in SMFs versus neurons.

For SMFs, the physiological “Na+-leak channel” (PNa) is not identified, but in neurons, smooth muscle, pancreatic cells, and others, the physiological (ion homeostatic) Na+-leak channel is a nonselective cation channel (Na leak channel nonselective [NALCN]; Lu et al., 2007; Kang et al., 2020). For decades, unidentified nonselective mdx fiber cation channels (their physiology unknown) have been considered problematic for passing unwanted ICa (Franco and Lansman, 1990; Yeung et al., 2005; Lansman, 2015; Ward et al., 2018). Whether they account for the SMF PNa is unknown, but as per Yeung et al. (2003) and as per simulations here, such channels, if overactive, would constitute a pathological INaleak.

DMD SMFs, whose ion homeostatic resilience declines with advancing disease state, experience what is, in effect, a chronic state of emergency. They cope by relying on the extraordinarily robust ion homeostatic strategy evolved by syncytial vertebrate SMFs for handling the inevitable (but for healthy fibers, transient) emergencies of SMF life: membrane tearing, interrupted vascular supply, and bouts of overexertion. Modeling suggests that refurbishing the emergency preparedness of DMD fibers (via improved operational pump strength and decreased Na+ leak via Nav1.4 and/or cation channels) could extend their already remarkable longevity.

A P-L/D model for SMFs

The features of SMF ion homeostasis are modeled using SM-CD, a zero-dimensional single compartment limited by a semipermeable membrane in an infinite (fixed concentrations) extracellular volume (Fig. 3 A, Table 2, and List of abbreviations). SM-CD constitutes one P-L/D ion homeostatic unit; multinucleate SMF would comprise hundreds or even thousands of such units (depending on sarcolemma area). A SMF with a smaller SA/Vol ratio than SM-CD would exhibit slower Δ[ion]i dynamics (Fig. 3, C and D). SM-CD encloses a fixed quantity of Donnan effectors: A, impermeant monovalent anions (as discussed by Fraser and Huang, 2004). The extracellular medium has a fixed A concentration. The membrane is permeable to Na+, K+, and Cl ions, whose extracellular concentrations are fixed. SM-CD has the same physical characteristics as the Dijkstra et al. (2016) neuronal cell, CN-CD (see Table 2): a resting cell volume (Volcell) of 2,000 µm3 and a constant total membrane capacitance of the cell model (Cm) corresponding to an SA of 2,000 µm2. As such, resting state cells are flaccid; lipid bilayers tolerate little lateral expansion, but only if a model cell inflated its membrane to spherical would tension and rupture be relevant (see Table 2). The permeation pathways of SM-CD and CN-CD (given below) include resting leak conductances (i.e., permeation pathways) for INaleak, IKleak, IClleak, and voltage-gated conductances (permeation pathways) for a transient sodium current, INav, for delayed rectifier potassium current, IKv, and CN-CD but not SM-CD has a Vm-dependent chloride current, IClv, plus an electroneutral cotransporter, JKCl. Driving forces acting on ions are, in all cases, electrodiffusive, as depicted by Goldman–Hodgkin–Katz (GHK) current equations (Hille 2001). The same 3Na+out/2K+in ATPase pump model used by Dijkstra et al. (2016) as taken by them from Hamada et al. (2003) is used throughout. It produces hyperpolarizing current,
INa,pump=3INaKpump=3/2 IKpump,
(1)
in response to the intracellular [Na+] (as per Fig. 4 A).

Since animal cells do not sustain osmotic pressures, intra/extracellular (osmolyte) inequalities elicit a H2O flow until osmotic balance is restored; this results in Volcell changes at rates limited by the slower net flux component, at any time, of the two ions ([Na+ + Cl]) whose joint net entry underlies osmotic Na+ loading.

Choice of leak permeabilities

Whereas CN-CD is precisely the Dijkstra et al. (2016) model, the SM-CD leak permeability ratio PNa:PK:PCl is broadly consistent with Fraser and Huang (2004); for more detail, see Setting Vrest in SM-CD and other CD models below.

GHK driving forces

Currents through open channels (permeability pathways), ion-specific or not, are modeled with the GHK formulation (Hille, 2001; see Fig. 4 C). For ionX = Na+, K+, or Cl, the GHK current is given by
IGHK(X,PionX,Vm)=PionX(zionXF) ×zionXFVmRT[ionX]i[ionX]eexpzionXFVmRT1expzionXFVmRT,
(2)
where PionX is the permeability, zionX is the valence, F = eNAvog is the Faraday constant (e is the electronic charge [1.6 × 10−19Coul] and NAvog is the Avogadro number), and [ionX]i and [ionX]e are the intra- and extracellular concentrations of ionX, respectively.

Resting or leak currents

Leak permeability mechanisms use the GHK formulation
IionX=I GHK(ionX,PionX,Vm),
(3)
where ionX is Na+, K+, or Cl. Leak permeability (PX) values were adjusted for model variants used here (Table 2) in the context of appropriate setting of Vrest.

Cation channel currents

For nonselective cation channels, we use a PK:PNa ratio of 1.11:1 and the formulation
Icatleakt=IGHKNa+,PNa,catleak,Vm+IGHKK+,PK,catleak,Vm.
(4)
In the present models, cation channel leaks do not contribute to healthy steady states; they are either transient stimulatory currents through SMF-endplate–type acetylcholine receptor (AChR) channels (Hille 2001), or pathological leaks (hence “leaky” cation channels).

Voltage-gated Na+ current

INav=m3hIGHK(Na+, PNav, Vm)
(5)

PNav is the maximal membrane permeability to Na+ through a V-gated channel (operating in a Hodgkin-Huxley [H-H] fashion). m is the H-H Na+ channel activation/deactivation gating variable, and h is the H-H Na+ channel inactivation/recovery gating variable. The current’s driving force also follows the GHK form of Eq. 1.

Delayed rectifier K+ current

IKv=n2IGHKK+, PKv, Vm
(6)

PKv is the maximal membrane permeability of K+ through a V-gated channel (delayed rectifier voltage-gated potassium channel [Kv], operating in a H-H fashion); n is the delayed rectifier K+ channel activation/deactivation gate variable.

Voltage-dependent gating

The nondimensional gating parameters m, h, and n evolve in time according to
dq/dt =αq(Vm)(1q)βq(Vm)q,
(7)
where q is m, n, or h, as defined above, and the voltage dependent αq(Vm) and βq(Vm) refer, for m and n, to gate activation and deactivation, and for h, to inactivation and recovery from inactivation. Table 3 gives the voltage dependences for the relevant rate constants.

Voltage-dependent Cl current

The SLC26A11 ion exchanger–based voltage-dependent Cl conductance (Rungta et al., 2015) is as described by Dijkstra et al. (2016):
IClv=IGHK(Cl ,PClv, Vm)1+exp(Vm+10 mV10 mV).
(8)
Except in CN-CD and minimal neuron (MN)–CD, this pathological CN-specific conductance is set at zero.

K/Cl cotransporter

This cotransporter (strength K/Cl cotransporter–maximal flux capacity [UKCl]) is present in CN-CD (SM lacks it; Pedersen et al., 2016) where (as per Dijkstra et al., 2016, except for an error in the log term of their Eq. 6) it is given by

JKCl=UKClRTFlnK+iCl-iK+eCl-e.
(9)

3Na+/2K+ ATPase pump current

The electrogenic 3Na+(out)/2K+(in)-ATPase modeled here (as in Dijkstra et al., 2016; and as plotted in Fig. 4 A) is given as
INaKpump([Na+]i)ImaxNaKpump=0.621+(6.7 mM[Na+]i)+0.381+(67.6 mM[Na+]i)3,
(10)
where
INaKpump=INa,pump+IK,pumporINaKpump=1/3(INa,pump)=1/2(IK,pump).
(11)
Thus, INaKpump signifies a net hyperpolarizing current (3Na+ outflow partially balanced by 2K+ inflow). This formulation is appropriate for models that assume an invariant extracellular medium. It depicts the pump’s two intracellular Na+-binding sites (with ∼10-fold different affinities) but has no term for the extracellular K+-binding site. Here, maximal pump strength (ImaxNaKpump) is varied in various computations (e.g., it is set to zero to depict anoxia or ouabain, diminished toward zero to depict lower percent pump strengths, and multiplied for up-regulation).

ATP consumption and hyperpolarizing INaKpump

ATP consumption by the 3Na+/2K+ ATPase pump in all the CD models is given as
ATP/s = INaKpump/F.
(12)
In other words, an INaKpump of 1 pA is equivalent to an ATP consumption of 10.38 amol/s.

Cell volume

Cellular swelling (or shrinking) is driven by an influx (or efflux) of water at rates that depend on the transmembrane osmotic gradient. The rate of change of cell volume, Volcell, due to water flux is given by
dVolcell/dt= JH2O=PH2OΔOsm,
(13)
where the osmotic gradient, ΔOsm = RT([Sol]i − [Sol]e), and [Sol]i and [Sol]e denote total concentrations of intra- and extracellular solutes, and PH2O is the effective membrane water permeability. This equilibration is typically (including in modeling here) assumed to be nearly instantaneous relative to the ion flows (Fraser and Huang, 2004; Dijkstra et al., 2016; Kay, 2017; though see also Dmitriev et al., 2019) so that osmotic ion fluxes, not H2O fluxes, are what limit the rate of osmotic swelling or shrinkage.

Nernst potentials

The equilibrium potential (Nernst potential) for each ion is
EionX=(RT/zionXF)log([ionX]e/[ionX]i),
(14)
where ionX is Na+, K+, or Cl ions, and zionX is the valence of each ion.

CD and membrane potential

Because CD models keep track of the absolute number of ions flowing across the cell membrane (of capacitance Cm), no differential equation is needed for Vm. Instead, an accounting equation is used, made simple because the extracellular space is kept neutral:
Vm=FCm1( NNa,i(t) + NK,i(t)NCl,i(t)NAi,i) = FCm1( dNNa(t)+ dNK(t)dNCl(t) dNAi),
(15)
where dNionX = NionX,j(t) NionX,i0 is the difference in the number of ions of species ionX, between its present value NionX,j and a reference value NionX,i0. The reference values are those yielding Vm = 0 mV, which is equivalent to a neutral intracellular space NNa,i0 + NK,i0 – NCl,i0 – NAi0 = 0. Because the quantity (attomol) of cytoplasmic impermeant anions (NAi) is constant, note that dNAi = 0 in Eq. 15. Number differences, dNionX, are easier to work with than total numbers of ions NionX (or than concentrations). For typical cell Cm values, voltages in the mV range correspond to net intracellular CDs (measured as a number of singly charged ions) of the order of amol (10−18 mol). For instance, when Vm varies in the range [−100 mV, 100 mV] this corresponds to a change in net singly charged ions of [−20.7, +20.7] amol. With our Volcell = 2,000 µm3, this yields tiny concentration changes [−0.01, 0.01] mM. For solutions whose concentrations are in the 1–100 mM range, there could be four or five orders of magnitude difference between steady-state values and the changes (in the 0.0001–0.01 mM range). Computations based on absolute concentrations, therefore, tend to be unstable. For this reason, in this study, calculations are based on changes in ion numbers (in amol units).

Number of intracellular ions

CD models account for the change in intracellular ions at any given moment (Fraser and Huang, 2004, 2007; Dijkstra et al., 2016), via the following simple relationships of the respective currents:

dNNa/dt =F1(INav+INaleak+3INaKpump),
(16)
dNK/dt =F1(IKv+IKleak2INaKpump)JKCl,
(17)
dNCl/dt=F1(IClv+IClleak)JKCl.
(18)

Setting Vrest in SM-CD and other CD models

An excitable cell’s Vrest (i.e., steady-state Vm) is typically more accessible (experimentally) than any cytoplasmic [ion] or Volcell. A consensus Vrest value is thus used for anchoring SM-CD; we chose Vrest = −86 mV. Other parameter determinants were then established iteratively as follows: first, a number is chosen for impermeant anions, NAi, consistent with the system’s total cation concentration (given the extracellular solution). Starting with CN-CD the value, we fine-tuned to meet our (self-imposed) requirement that SM-CD and CN-CD have the same resting Volcell. Thus, note in Table 2 the slightly different NAi for CN-CD and SM-CD (likewise, low ATPase [LA]–CD and SM-CD) and identical steady-state Volcell.

At steady state, Na+ and K+ leak currents (for a system with a given pump strength), must precisely balance. In other words, Vm converges (along with ion concentrations and Volcell) on steady state (=Vrest) when
passive Na+ influx+active Na+ extrusion=zero
and
passive K+ efflux+active K+ uptake = zero.
For a 3Na+/2K+ ATPase, this steady-state requirement is met when
INaleak=INa,pump=3/2(IK,pump)=3/2(IKleak).
(19)
For P-L/D systems of a given pump strength, Vrest varies monotonically with PNa:PK (Fig. 4 D); low ratios yield hyperpolarized Vrest values, high ratios, depolarized ones. Vrest = −86 mV for SM-CD (and WD-CD) requires PNa:PK = 0.03:1. Absolute values for PNa and PK, and for PCl (Table 2), were guided by the PNa:PK:PCl ratio (0.02:1:3) reported for amphibian SM (Fraser and Huang 2004) and our choice to give SM-CM the same area (Cm) as CN-CD.

As per Eq. 16, a pump stoichiometry other than 3Na+out/2K+in would, all else being equal, alter Vrest. Pump stoichiometry is invariant here, but see Dmitriev et al. (2019).

Excitability and safety factor for SM-CD

For SM-CD to be hyperpolarized and appropriately excitable (i.e., relatively inexcitable), its input impedance had to be (1) notably less than for CN-CD and (2) predominantly PCl-based (Pedersen et al., 2016). With resting P values set, the need to trigger spikes near −60 mV (Fu et al., 2011) with a reasonable-sized safety factor (Ruff 2011) had to be met. To achieve safety factor ∼1.5, Nav and Kv densities in SM-CD were set at 3× the CN-CD level (a larger PCl would have required even greater V-gated channel densities). Thus, absolute PCl, PNa, and PK values of SM-CD are biologically appropriate, but leave room for physiological modulation (to, say, alter Vrest via ΔPNa or ΔPK, or to modulate excitability by ΔPCl).

Table 2 shows that m3h is vanishingly small at Vrest in SM-CD; for CN-CD it adds an extremely small Nav channel contribution to the operational value of PNa that negligibly affects Vrest.

Cytoplasmic Donnan effectors

Once Vrest is set via PNa and PK, the intracellular anion concentrations are determined uniquely for the resting state. If Cl transport is purely passive (i.e., no involvement of secondary transport) as in SM-CD, then ECl = Vrest, and therefore,
[Cl]i=[Cl]eexpFVrestRT.
(20)
The value of [A]i then follows from the voltage and osmotic balance requirements.
The voltage requirement yields
[Na+]i+[K+]i[Cl]i[A]i=Δconc,
(21)
where Δconc is the tiny excess concentration of anions associated with Vrest and equal to [intracellular anions] − [intracellular cations]
Δconc=VrestCmFVolcell.
(22)
For Vrest = −86 mV, Δconc = 0.00886 mM. The osmotic equilibrium condition is
[Na+]i+[K+]i+[Cl]i+[A]i=[S]i=[S]e,
(23)
where [S]i and [S]e are defined below Eq. 13. Eqs. 21 and 23 yield
Ai=Si+Δconc/2Cli.
(24)
A P-L/D model-cell’s design for steady state imposes its [A]i. Therefore, the choice of NAi determines the cell volume Volcell = NAi/[A]i. As mentioned above, SM-CD’s NAi was set to give a resting Volcell = 2,000 µm3. Here, the quantity of impermeant anions NAi is invariant, but for many in vivo circumstances, it would vary, and Volcell would vary accordingly in such cases.

Instantaneous perturbations

Experimental solution changes (e.g., as in brain slice experiments) typically require finite “wash-in/wash-out” times. Dijkstra et al. (2016) mimicked such solution changes (affecting pump rates and channel gating, etc.), but here, doing so would have unnecessarily obscured mechanistic underpinnings of responses. Thus, pump-off (anoxia) and pump-on (restoration of pump strength) changes and channel gating changes (Nav and cation channels open probabilities) are applied instantaneously.

Maximum cell volume before lysis

Both CN-CD and SM-CD have Cm = 20 pF and steady-state Volcell = 2,000 µm3. If 0.01 F/m2 (= 0.01 pF/µm2) is the specific capacitance of the bilayer, membrane area is 20/0.01 = 2,000 µm2. With 4πR3/3 the volume of a spherical cell and 4πR2 its surface area (SA), maximum Volcell as the cell swells (to spherical) would be = 4/3π(2,000/4π)3/2 = 8,410.4 µm3. Given a 4% bilayer elasticity strain limit (yielding membrane area = 2,080 µm2), rupture would occur at 8,920 µm3. Thus, in bifurcation plots, the notional DE values indicated for reference are unachievable by these models. Note too that present models depict neither SA regulation nor membrane tension homeostasis (see Morris, 2018).

Excitatory post-synaptic current (EPSC) via AChR channels

SM-CD APs are initiated by macroscopic EPSC through AChRs, which are nonselective cation channels that pass Na+ and K+ as per the GHK formalism. Our EPSC time course mimics a g(t) reported by Wang et al. (2004). As per Hille (2001), PK:PNa for IEPSC = 1.11:1. The function g(t) has a maximum of 1, and PNa,EPSC yields a 1.5-fold safety factor (i.e., an amplitude adjusted to 1.5× the threshold required to elicit an AP in SM-CD).

The end-plate current is

IEPSC(t)=g(t) [IGHK(Na+, PNa,EPSC, Vm)]+[IGHK(K+, PK,EPSC, Vm)].
(25)

Nav–coupled left-shift (CLS) depiction of bleb damage to Nav-bearing membrane

The extent (in mV) of Nav-CLS as depicted in Fig. 1 D increases with bleb-damage intensity (Wang et al., 2009; Boucher et al., 2012; Morris et al., 2012a, 2012b; Joos et al., 2017). Cell survival seems improbable with 100% of Nav-bearing membrane damaged so arbitrarily. We model damage to 30% of the Nav population (i.e., affected channels [AC] = 0.3); imposed damage intensity is thus Nav-CLS(0.3) mV. Voltage dependences of rate constants αq (V) and βq (V) for m and h are the same as in Eq. 7, but shifted in the hyperpolarizing direction by left shift (LS; in mV). Therefore, for the affected Nav channels (AC), m and h evolve according to
dq/dt=αq(Vm+LS)(1q)βq(Vm+LS),
(26)
where q is either m or h.

Computational methods

Calculations involved solving sets of first order differential equations. These were done using Python with the ordinary differential equation solver odeint.

Online supplemental material

Fig. S1 is a high-resolution look at P-L/D processes as they move away, then back to steady state, during the several minutes it takes to redress the ion perturbations associated with a single synaptically triggered AP. Note the question and answer section in the supplemental text; it pertains to Fig. S1. Fig. S2 shows anoxic rundown (Vm(t) only) for SM-CD at variable time resolutions; anoxic rundown trajectories for CN-CD (Vm, [ion]i, Volcell); anoxic rundown for WD-CD (Vm and ATP consumption); an example of the consequences of altering pump Michaelis–Menten constants (dose–responses and steady-state [Na+]i computed for SM-CD in each case); and for the ischemic Vm(t) rundown of Fig. 6, the concurrent [Na+]i(t). Fig. S3 shows multiparameter bifurcation plots for SM-CD, including the spontaneous Vm(t) trajectory from the pathological steady-state continuum back to the continuum of physiological steady states. Table S1 emphasizes that, while in humans, the large disparity between the tissue mass of brains and SM is well-recognized, in the common ancestors of contemporary vertebrates, this disparity would have been substantially greater, making costly neuronal ion homeostasis in those early vertebrates relatively unproblematic, while placing a premium, even then, on the evolution of efficient and robust SM ion homeostasis. Supplemental text at the end of the PDF includes sections entitled P-L/D modeling of myotonia congenita: SMFs with a [small PCl][small INaleak] steady state, and P-L/D modeling of SM injury, tourniquets, compartment syndrome.

P-L/D systems at steady state

Fig. 3 A and Table 2 show how SM-CD, a generic P-L/D model for excitable SMF membrane, parallels CN-CD (see Materials and methods). SM-CD uses the minimal set of flux elements (Fig. 2) needed for autonomous return to ion homeostatic steady state after an ionic perturbation. CN-CD incorporates a K/Cl cotransporter and thus has a nonminimal P-L/D steady state (poisoning the K/Cl cotransporter reveals its impact; in Table 2, compare steady-state values for CN-CD versus MN-CD). Including SMFs’ many cotransporters (see Fraser and Huang, 2004; Usher-Smith et al., 2009) in generic SM-CD would have been unhelpful (nevertheless, the CN-CD/MN-CD exercise is a simple how to). Because the extracellular milieu is fixed, pump sensitivity to [Na+]i (see Materials and methods; Fig. 4, A and B; Eq. 10) but not to [K+]ext is invoked.

At ion homeostatic steady state, by definition, passive entry and active extrusion of Na+ and of K+ precisely balance. In all models, PK >> PNa, and electrodiffusion driving forces (see Fig. 4 C) are almost always greater on Na+ than on K+. As detailed in Materials and methodsand shown in Fig. 4 D, the PK:PNa ratio sets Vrest in conjunction with pumping that always operates electrogenically (3Na+out/2K+in). Consequently, both the passive and active fluxes of K+ are constrained by the passive and (hyperpolarizing) active fluxes of Na+. Accordingly, when steady state is perturbed, INaleak (not IKleak) is rate limiting for flux trajectories, so throughout, Na+ fluxes are emphasized while the attendant K+ fluxes, though plotted, are mentioned less frequently.

In neuronal CN-CD, with PK the major permeability and resting INaleak large (78.2/14.75 = 5.3× that of SM-CD; Table 2, steady-state section), there is a Pump-Leak–dominated steady state, meaning that the balanced fluxes mostly use PK, PNa, and the Na+/K+ pump. For SM-CD, with PCl the major permeability and resting INaleak very small, steady-state fluxes are mostly permeant anions (only Cl is modeled) via PCl whose influx/efflux is balanced by anionic Donnan effectors. SM-CD has a Donnan dominated steady-state. Before making further inter-model comparisons and subjecting them to DMD-like deficits, SM-CD’s broad “SMF credibility” is assessed via a physiological stress test.

Stress testing SM-CD

The Fig. 5 A AP stress test mimics a procedure for isolated rat soleus muscle, except that SM-CD is stimulated not electrotonically but by a train of EPSCs (through cation channels; see Materials and methods). As APs fire, [Na+]i rises, [K+]i falls, and ATP consumption abruptly increases. This is the P-L/D system’s [Na+]i-sensitive P-L active feedback in operation. Simultaneously, small increases in [Cl]i and Volcell reflect the Donnan effector–mediated passive feedbacks (Fig. S1 details the EPSC and AP fluxes).

During, between, and after APs, permeability and driving forces for K+ and Na+ almost match. With the constraint for compartment electroneutrality thus almost met, the small excess Na+ influx is addressed by Cl+ H2O influxes. Immediately after APs stop (AChR cation channels, Nav, and Kv all closed), only elements of SM-CD’s minimal P-L/D system remain in play. Vm hyperpolarization reflects the ↑[Na+]i, as pumping ↓[Na+]i to 3.7 mM, Vm converges to Vrest (INaleak = −INaKpump).

During the autonomous return to steady state, [Na+]i and [K+]i change monotonically, but [Cl]i and Volcell oscillate. The quantity NAi (attomol of cytoplasmic Donnan effectors) is fixed; consequently, [A]i falls and rises inversely with Volcell and [Cl]i (not shown). During the AP train, [Cl]i increases slightly with each AP’s small excess Na+ influx and consequent electro-neutralizing Cl influx, a double entry constituting a cyto-osmolyte excess. With H2O activity higher externally than internally, an osmo-balancing influx of H2O occurs. Thus, for minimal SM-CD, [Cl]i changes (∼4 mM increase when APs stop) are precisely mirrored by Volcell changes (∼2.5% increase; ΔVolcell <50 µm3; because PH2O is very large, no time lag is evident). Were the system to swell to notional DE (Table 2), Volcell (initially 2,000 µm3) would be ∼14,000 µm3 (↑700%). This provides a “thermodynamic size gauge” of how effectively, during 1,200 EPSC-triggered APs, Donnan dominated SM-CD forestalls osmotic swelling. With [INaleak] so small, [big PCl] does not render SM-CD vulnerable to osmotic Na+ loading. In SM-CD, the driving force on Cl approaches 0 as the system converges on Vrest (a minimal P-L/D steady-state feature).

At 120 Hz, ion homeostasis does not quite fully restore steady state before the next AP, so parameter changes mount. When firing stops, reshrinkage begins immediately, undershooting before Volcell reconverges on steady state. This [Cl]i(t), Volcell(t), [A]i(t) oscillation, or “Donnan bounce,” is the slowest aspect of ion homeostatic restoration. Normal excitability would seldom be as demanding as the stress test; the speedy system rebound (only slightly more prolonged after 1,200 APs than after 1 AP (as per Fig. S1) is consistent with a hefty pump-reserve capacity for handling additional ENa-depleting tasks.

To handle multiple gradient-dissipating tasks (e.g., APs, ENa-dependent secondary transport, restoring gradients after microtear repair), ion homeostatic systems need a reserve, i.e., pump strength in excess of the steady-state requirement. Pump reserve is the thus the ratio (maximal ATP consumption)/(steady-state ATP consumption); or, in electrophysiological terms, pump reserve is (maximal INaKpump)/(steady-state INaKpump [= −steady-state INaleak]). The SM-CD pump reserve is 11.1-fold (see Table 2); for various rat muscles, pump reserves in the range 7–22-fold are reported (Clausen, 2015).

In DMD, pump reserve would decrease as pump strength fell and/or as Na+ leaks increased. Though maximal (100%) pump strength is identical in healthy CN-CD and healthy SM-CD, the [big INaleak] of CN-CD leaves it with a mere 2.1-fold pump reserve. No one parameter expresses how pump strength relates to a system’s global physiological resilience, but SMFs’ large pump reserves augur well.

For the same stress test but with pump reserve doubled (pump strength up-regulated to 200%), Fig. 5 B plots ATP consumption; peak ATP consumption is greater and recovery is faster (see legend regarding the recovery in rat soleus with a similar pump reserve) and (not shown) steady-state [Na+]i would drop to 2.7 mM (from 3.7 mM). The steady-state ATP consumption increase is trivial (compare Fig. 5, A and B, baselines: the ATP consumption increase in Fig. 5 B is almost undetectable). This feature (the ability to increase pump reserve with almost no increased steady-state expenditure) is crucial for physiological resilience. But to respond to transient physiological Na+ loading, mechanisms other than boosted pump strength are invoked; during episodes of intense firing, SMFs have a rapid-acting (seconds) “excitation-activation feed-forward” process that results in sustained post–AP-train [Na+]i undershoots (Nielsen and Clausen, 1997). While pump up-regulation (as per Fig. 5 B) operates in the direction needed, it would be too slow. The data by Nielsen and Clausen (1997) point to more expeditious mechanisms involving altered transport characteristics. For instance, if excitation-activation feed-forward signaling were to act by (arbitrarily) halving the SM-CD pump’s Na+-binding Michaelis–Menten constants, a [Na+]i undershoot (3.7→1.9 mM) would result (see Fig. S2 E). For perspective, pump strength up-regulation to 1,000% (ImaxNaKpump to 545 pA; maximal ATP consumption to 5,660 amol/s) reduces steady-state [Na+]i to 1.7 mM. While increased quantities of functional pump protein (thence bigger pump reserves) serve overall resilience well, fast-acting (and presumably temporary) pump kinetic adjustments serve physiological agility.

In summary, SM-CD, though radically simple, handles an excitability stress test in a manner qualitatively similar to rodent fibers. This general verisimilitude justifies using and modifying SM-CD to learn how altered or added features reflect the DMD situation.

Syncytial SMF morphology boosts ion homeostatic robustness

Steady state P-L/D values depend on membrane SA. Time courses depend on SA/Vol (represented here as Cm/Volcell). All models here have SA = 2,000 µm2 enclosing steady-state Volcell 2,000 µm3 (Table 2; and Fig. 3, A and B). However, to relate SM-CD to particular myofibers, syncytial morphology (Fig. 3, C and D) would matter. In an SMF, each 2,000 µm2 unit would encircle (not enclose) a myoplasmic volume. One fiber would comprise hundreds to thousands of contiguous 2,000 µm2 SM-CD ion homeostatic units, with cylindrical slice width varying with fiber radius (Fig. 3 D, i). Steady state would be 10× more efficient in a myofiber with SA/Vol 0.1 that of SM-CD. During rundown, that myofiber’s gradients would benefit from slower passive dissipation than SM-CD, but once dissipated, active recovery would also be slower than in SM-CD.

Steady-state costs: CN-CD versus Donnan dominated SM-CD versus Pump-Leak–dominated WD-CD

The PNa:PK ratio of SM-CD puts its Vrest >20 mV more negative than the reference model, CN-CD. In spite of the bigger driving force on Na+, SM-CD’s extremely small PNa makes its [small INaleak] and thence its steady-state ATP consumption 5.3× smaller. Since all models have the same maximal (100%) pump strengths, SM-CD’s pump reserve, too, is 5.3× that of CN-CD (Table 2). SM-CD pump reserve would coincide with CN-CD’s meager normal (2.1-fold) value when SM-CD pump strength fell to a mere 18%. Thus, even without factoring in syncytial morphology (a feature of SMFs, and not of neurons), the 5.3× differential bespeaks the extraordinary frugality and robustness of the ion homeostatic strategy adopted by SMFs relative to that of central neurons.

SM-CD/CN-CD comparisons are physio/pathophysiologically informative. They do not, however, provide a direct readout of how much of SM-CD’s robustness is attributable to its Donnan dominated steady state. For this we devised a counterfactual (Pump-Leak dominated) SM-CD analog, WD-CD. WD-CD, like SM-CD, is a minimal P-L/D system. Its maximal pump strength, PNa:PK ratio, Vrest, and low input impedance are all identical to SM-CD’s (Table 2). Having set the WD-CD PCl equal to CN-CD’s, we matched SM-CD’s low input impedance via large-valued PK and PNa (in Table 2, see absolute permeabilities and PNa:PK:PCl). WD-CD has [small PCl][big INaleak] steady state that consumes ATP at 180 amol/s (versus 51 amol/s for SM-CD). This drops WD-CD’s pump reserve to 3.1-fold (SM-CD: 11.1-fold). While WD-CD can handle the stress test, it is continually consuming (180/51) 3.5× more ATP than SM-CD, and after stress test, it takes more than two times longer than SM-CD to restore steady state (Fig. S2 D versus Fig. 5 A). For energetic efficiency, and by extension, for resilience during emergencies, SM-CD far outmatches its electrically equivalent analog, WD-CD.

Provenance of DMD fiber Na+ overload: Insufficient pumping? Too much leaking?

The provenance of chronic DMD fiber Na+ overload is unclear. To address this via SM-CD, we therefore ask the following: could chronic (i.e., steady state) overloads arise solely from (1) too little Na+ pumping (low pump strength) or from (2) too much Na+ leaking (leaky channels in damaged sarcolemma), or (3) do both contribute? As a principal cause of chronic DMD Na+ overload (i.e., with pump strength at 100% and Na+-permeant channels all functioning normally), hyperactive secondary transporters are not plausible and are not addressed here.

The magnitude of PNa in DMD fibers is unknown since SMF PNa is not identified. But mdx fibers have leaky nonselective cation channels (identified and unidentified; Carlson and Officer 1996; Lansman, 2015) and leaky Nav1.4 channels (Hirn et al., 2008). SMF pump proteins are well studied (e.g., Clausen, 2013, 2015; Hakimjavadi et al., 2018; Kravtsova et al., 2020), but for this generic comparison of SMF versus neuronal ion homeostatic strategies, we kept the same simple pump model as in CN-CD.

Kravtsova et al. (2020) report diminished pump-protein efficacy in mdx fibers, and several studies addressing Na+-overload point to diminished operational pump strength. Modulators that depress pumping (a machinery issue) increase mdx fiber Na+ overload (Table 1, item 4; Miles et al., 2011), and stimulating NO pathways (a supply issue related to functional ischemia) almost fully abolishes Na+ overload (Altamirano et al., 2014; Table 1, item 7). After reporting chronically elevated Na+ in DMD patients’ muscle (Table 1, item 5), Lehmann-Horn et al. (2012) noted unexpectedly improved muscle function in a dystrophic patient treated (to alleviate tissue edema) with eplerenone. Probing the mechanism via a rat diaphragm DMD model (Breitenbach et al., 2016), they found that eplerenone up-regulates Na+/K+-ATPase (via α-subunit Tyr10 dephosphorylation), causing an ouabain-sensitive fiber repolarization.

To exemplify an advanced-DMD (i.e., after infancy; Fig. 1 A) SM-CD variant, we use LA-CD (Table 2): it is SM-CD with pump strength at 30% and NAi decreased slightly so Volcell = 2,000 µm3. Further, we systematically characterize SM-CD’s P-L/D characteristics across pump strengths (SM-CD steady states as pump strength varies, below).

Pump reserve in DMD-like LA-CD is 3.4-fold (better than 2.1-fold in healthy CN-CD, substantially less than the 11.1-fold of healthy SM-CD), but Vrest and ATP consumption for LA-CD and SM-CD are almost the same. LA-CD’s 6.5 mM [Na+]i is inside the healthy range for mouse fibers according to some studies (see Table 1). So would LA-CD really represent an ailing DMD-like fiber? Yes. [Na+]i values for mice are extremely variable, but if, say, an mdx mouse fiber was using an ENa-depleting Na+ transporter (e.g., Iwata et al., 2007), 6.5 mM could easily become 7.3 mM, which Burr et al. (2014) (Table 1, item 8) report as chronically Na+ overloaded for mdx fibers. Moreover, even without knowing that healthy rodent SMFs maintain pump reserves in the 7–22-fold range (versus LA-CD’s 3.4-fold), LA-CD is a substandard P-L/D system vis-à-vis steady-state energetics: whereas SM-CD consumes ATP at 51 amol/s to achieve [Na+]i = 3.7 mM, LA-CD consumes ATP at 50.7 amol/s to achieve only 6.5 mM (see Materials and methods; Fig. 4 B, ii). LA-CD’s minimally lower ATP consumption is clearly no bargain. Compared with SM-CD, LA-CD would be classified as chronically Na+ overloaded, due solely to too little pumping. How nonlinear aspects of Na+ fluxes (too little pumping and too much leaking) contribute here will emerge further in the systematic pump-strength analysis (below).

Ischemic rundown to spontaneous firing

DMD fibers suffer exertion-induced bouts of functional ischemia (Thomas, 2013). A downstep to 0% pump strength would mimic anoxic or ouabain-poisoned rundown. As if severe ischemia suddenly reduced the ATP supply, Fig. 6 follows Vm(t) in SM-CD after a pump-strength downstep from 100% to 7.9% (maximally 44.7 amol/s ATP consumption, or 4.3 pA of INaKpump; trial and error show that 8% suffices but 7.9% is marginally less what SM-CD requires to maintain a steady state). In this trajectory, the system seems to restabilize near −65 mV but in fact continues depolarizing for hours at an exquisitely slow rate (barring any current noise, a syncytial fiber would depolarize even more slowly). Then, spontaneously, it fires APs. Why? Because at that Vm, INaleak (through PNa + Nav channels; in Fig. 4 E, see m3h(Vm)) exceeds the maximal Na+ extrusion achievable at 7.9% pump strength (4.3 pA). When firing ceases (after ∼30 s), the system slowly converges on a profoundly depolarized, pathological Vrest. As the inset plot shows, with even deeper ischemic downsteps, rundown to spontaneous firing speeds up, taking 22.7 min for 100%→0% pump strength (Fig. 6 plots only Vm(t), but [Na+]i(t) is in Fig. S2 F).

Fig. 6 could mimic application of tourniquets (MacDonald et al., 2021) and compartment syndrome (Tatman et al., 2020). Depending on the severity of vascular constriction in compartment syndrome, ischemic and anoxic fibers’ rundown to firing threshold would vary enormously, consistent with reports, in compartment syndrome, of the unpredictable timing of lethal threshold events (Johnstone and Ball, 2019). In DMD, compartment syndrome is thought to contribute to limb contracture (Siegel, 1992; Dooley and Chiasson, 2014). Extremely slow ischemic rundown buys time, fostering survival in connection with bouts of functional ischemia.

Anoxic rundown to spontaneous firing

In Fig. 6, pump feedback continually fights the passive cation (Na+ and K, w) leaks (though too weakly) while, concurrently, Donnan effector–mediated feedbacks operate at full force. In anoxic rundowns (i.e., 100%→0% pump strength at t = 0; Fig. 7), only the passive processes remain operative. Fig. 7 A, i, shows the SM-CD trajectories during anoxic rundown; spontaneous firing starts at 22.7 min. This Vm(t) is expanded in Fig. S2 A and compared there (Fig. S2 B) against the Vm(t) for CN-CD, where spontaneous firing starts at ∼100 ms. Because Dijkstra et al. (2016) mimic a brain-slice experiment’s slow wash-in of the pump-poison ouabain, their rundown is slow. Step changes, as used here, render the biophysics more transparent (e.g., INaKpump-off for SM-CD causes a ~Δ2 mV RC-type depolarization).

During prefiring rundown in SM-CD, Na+ influx (small PNa, large initial driving force) and K+ efflux (large PK, small initial driving force) nearly match. But a small excess (net) Na+ influx engenders small net influxes of Cl and H2O (ΔVolcell reflects a net H2O flux); electro-neutrality and osmo-balance are thus maintained (see expanded prefiring Cl and Volcell trajectories, Fig. 7 A, ii).

SM-CD’s biggest leak, PCl, does not set the pace for rundown toward DE. If, at t = 0, PNa and PK were blocked along with the pump, the cytoplasmic Donnan effectors would hold SM-CD at Vm = ECl = −86 mV. Upon reopening of PNa and PK, rundown would commence; the system’s smallest leak ([small INaleak]) with its large driving force would set the rate for ion gradient dissipation. Note that SM-CD depolarizes from −75 mV to −65 mV in ∼12 min, reasonably close to the rundown rate of ouabain-poisoned rat soleus SMFs (10 mV/10 min; Clausen and Flatman, 1977), whose rundown is consistent with a somewhat larger operational PNa and with those fibers’ more depolarized Vrest. Fig. 7 B, i, shows how sensitively SM-CD responds to a ΔPNa; there a 27% ↓PNa (0.3→0.22 µm3/s) hyperpolarizes Vrest (−86→ −90 mV), reduces steady-state ATP consumption (51→39 amol/s), and prolongs prefiring anoxic rundown (22.7→34.0 min).

With spontaneous firing, system permeabilities change vastly; intermittently open Nav and Kv channels support very large Na+ influxes and K+ effluxes. Cl influx through SM-CD’s [big PCl] increases substantially to neutralize the now very large excess Na+ influx. With that [Na+ + Cl] comes osmo-balancing H2O; spontaneous firing at 0% pump strength causes gross inflation, i.e., worst case scenario cytotoxic swelling.

Does PCl influence prefiring rundown speed? Negligibly in SM-CD, because always, the electro-diffusion driving force acting on Cl is near zero. Thus, for 0.1× PCl, 1.0× PCl, and 10× PCl, spontaneous firing starts at 20.7, 22.7, and 23.8 min, respectively (not shown). Throughout prefiring rundown, SM-CD’s [big PCl] supports large Cl effluxes and (marginally larger) influxes, but while net Na+ influx stays small, net Cl influx stays equally small. The 7.9% pump-strength rundown of Fig. 6 is so slow because, there, pumping almost (but not quite) counters that small net leak. An implication is that though ↑PCl strongly inhibits endplate-triggered APs under energy-depleted conditions (Bækgaard Nielsen et al., 2017; Leermakers et al., 2020), it would not help forestall spontaneous firing in fibers whose pump strength was too low to sustain a steady state. Under those same conditions, however, a ↓PNa (as per Fig. 7 B, i) would both diminish endplate excitability (via Vrest hyperpolarization) and forestall spontaneous firing (slower rundown).

For SMFs in low pump-strength scenarios like Fig. 6 and Fig. 7 A, excitation–contraction coupling triggered as spontaneous firing started could result in fiber-destroying contractures (due release of [Ca2+]i; Claflin and Brooks, 2008). Though ion homeostatic failure would have brought on fiber demise, it might be characterized as Ca2+ necrosis.

Ion homeostatic recovery and spontaneous firing

The pathophysiological virtue of very slow rundown is evident in Fig. 7 A, iii: if pump strength is restored to 100% at any pre-time firing, the system can return to steady state. Just before firing starts, with [Na+]i at 20× its steady-state level, the resumed [Na+]i-sensitive Na+-extrusion (INaKpump) causes Vm to hyperpolarize until the system reconverges to steady-state [Na+]i and Vrest. If pumping is restored during spontaneous firing, the system can still return to steady state, but in situ, prospects for such recovery would be moot if anoxic-condition contractures (triggered by the spontaneous APs) had destroyed the fiber (Claflin and Brooks, 2008).

A pathological P-L/D steady state: Nav window conductance

If pump strength is restored even 100 ms after spontaneous firing stops (Fig. 7 A, iii, bottom), the system shows new behavior. Unable to return to healthy Vrest, SM-CD instead converges on a depolarized ATP-devouring steady state of degraded ion gradients and a pathological Vrest (−22 mV). Hyperpolarizing INaKpump, though maximally stimulated by the extreme [Na+]i, cannot surmount the depolarizing INaleak. The problem: operational PNa has acquired a new component (i.e., one not in force at SM-CD’s healthy steady state). In the pathologically restabilized system, Nav channel window conductance contributes to INaleak (for m3h(Vm) at −22 mV, see Fig. 4 E). The system is in depolarizing block (not shown). To return from this pathological steady state to a physiological steady state would require that maximal INaKpump up-regulate enough that hyperpolarizing INaKpump would exceed the depolarizing INav-augmented INaleak (≥339% pump strength is needed; Fig. S3 D, i and ii). For CNs, Dijkstra et al. (2016) propose such recovery scenarios; whether comparable scenarios relate to SMF ischemia-reperfusion injury is outside our scope (Dudley et al., 2006; Schmucker et al., 2015; Li et al., 2020).

The DMD-like low pump-strength SM-CD variant (LA-CD) can handle rundown and recovery

Chronically low–pump-strength (30%) LA-CD has the same PNa as SM-CD. Though LA-CD is mildly Na+ overloaded, its small PNa (thence [small INaleak]) keeps anoxic rundown (Fig. 7 C, i) almost as slow as for SM-CD; it fires spontaneously at 22.0 min (22.7 for SM-CD), passing currents essentially indistinguishable from SM-CD (shown for LA-CD only; Fig. 7 C, ii). LA-CD’s pump reserve (3.4-fold) suffices to restore the system at any point during rundown and until spontaneous firing stops (Fig. 7 C, iii), but recovery is slower than for SM-CD.

Donnan-driven swelling: Small PNa versus small PCl

A priori, if a P-L/D system’s only flux mechanisms are those operative at steady state, temporary loss of pump strength is always fixable (ignoring perils, in situ, from extreme cell inflation). Thus, systems with V-gated channels zeroed recover after prolonged, deeply depolarizing anoxia, as shown (Fig. 7 D) for SM*-CD (SM-CD with V-gated channels eliminated). After 105 min of anoxia, Vm has depolarized to −20 mV. Then, ∼5 min after restoration of 100% pump strength, Vm = −86 mV; SM*-CD recovers fully (albeit more slowly) with restoration to any value ≥8% pump strength (not shown).

For all parameters, Fig. 8 compares rundown trajectories for SM*-CD and CN*-CD (CN-CD without V-gated channels). Note how SM*-CD, which has [big PCl] and small PNa, exerts a more powerful brake against Donnan effector–induced swelling than CN*-CD, which has [small PCl] and large PNa. The [big PCl]/(Donnan effector) collaboration used by SMFs lets them be electrically leaky yet metabolically tight.

Fig. 8 suggests that during prolonged quiescent periods (as different as hibernation or post-injury tissue remodeling; Jackson, 2002; Baumann et al., 2020), SMFs could optimize prospects for eventual ion homeostatic recovery by preemptively zeroing Nav1.4 channels (e.g., by promoting slow-inactivated [Webb et al., 2009] or other nonpermeant states [Kiss et al., 2014]; likewise for ↓PNa and ↓PK [Donohoe et al., 2000]).

SM-CD steady states as pump strength varies

Generic neuron models (Hübel et al., 2014) including CN-CD (Dijkstra et al., 2016) show that ion homeostatic steady states change nonlinearly as pump strength falls. Spontaneously, at discrete points (thresholds) in parameter space, CN-CD destabilizes and exhibits bistability. These characteristics are also evident in SM-CD. P-L/D bistability is unrelated to whether a system’s steady state is dominated by (actively balanced) cation fluxes or (passively balanced) anion fluxes; bistable ion homeostasis is a (computational) trait of P-L/D systems with embedded voltage-gated channels.

SM-CD has two nonlinear Na+ flux mechanisms: INaKpump([Na+]i) and INav(Vm). How, in principle, their interplay renders SM-CD computationally bistable is shown with the steady-state Vm (pump strength) bifurcation plot in Fig. 9 A (the complete set of SM-CD bifurcation plots, plus the Vm(t) recovery trajectory at 339% pump strength, is in Fig. S3). The solid black line is the continuum of physiological steady states. Spanning the same pump-strength range, the solid blue line is a continuum of pathological steady states. “Bistable” signifies that, across part of the pump-strength range, these two continua overlap. On each continuum, an unstable threshold (bifurcation point) occurs: X and # at pump strengths 7.9% and 339%, respectively. For CN-CD, analogous instabilities (X and #) occur at pump strengths 65% and 181%, respectively (Dijkstra et al., 2016).

For SM-CD, the expectation (in situ) of lethal contracture at X (7.9% pump strength) makes the upper continuum moot. SM-CD is stepped from 100% to 7.9% pump strength in Fig. 6, but if SM-CD was stepped (say) 8% pump strength→X (7.9%), spontaneous firing and (in situ) contracture would follow almost immediately.

Assuming the pathological steady-state continuum to be inaccessible in situ, Fig. 9 B plots only physiological steady states, and only for the range most relevant to DMD (encircled in Fig. 9 A), i.e., from 100% → through the saddle-node (X) to 0% pump strength. SM-CD, 30% pump strength, plotted here is almost the same as LA-CD.

LA-CD is the tolerably robust low pump-strength (30%) system (e.g., Fig. 7 C) whose elevated steady-state [Na+]i already indicated that, in principle, too little pumping alone could underlie a small chronic Na+ overload. The Fig. 9 B [Na+]i plot shows that as pump strength drops below that 30%, extreme chronic Na+-overload values develop, approaching 88 mM before system destabilization at the saddle node (7.9% pump strength, X). At pump strengths from ∼20% to 15% down through 8%, SM-CD would unquestionably be considered chronically Na+ overloaded.

Insofar as pump strengths ≤30% mimic advanced disease-state DMD fibers, chronic Na+ overload of DMD/mdx fibers could result solely from too little Na+ pumping with the proviso that, in that situation, unwanted Nav channel window conductance increases operational PNa in the low pump-strength danger zone leading to X. That pathological ↑PNa, we emphasize, is attributable to normally functioning Nav channels. Nav channels overactive in the damaged sarcolemma of DMD fibers will be addressed below.

Functional ischemia: Moving in and out of the danger zone

In DMD, transient functional ischemia could take chronically low pump-strength fibers (say, at 30%) perilously rightward toward X, as depicted in Fig. 9 C,. Equally important, however, is what this plot conveys regarding prospects for recovery as fibers approach X: 8% pump strength is still on the physiological continuum, so any intervention, no matter how small, that elevates operational pump strength >8% will contribute to fiber survival by moving the system leftward. This seems self-evident/unremarkable until consideration is given to CN-CD, whose saddle node (X) is at 65% pump strength. If CN-CD pump strength falls <65%, recovery is only possible if pump strength can be boosted to ≥181% (yielding the ↑hyperpolarizing INaKpump needed to take CN-CD to its #, the unstable recovery threshold on its pathological continuum; Dijkstra et al., 2016; or pink X, # in Fig. S3). For DMD-afflicted SMFs, Fig. 9 C suggests that provided pump strength is ≥8%, even procedures as minor as massage treatments and vibration that improves bloodflow could, and evidently do (Saxena et al., 2013; Carroll et al., 2020), improve DMD fiber viability.

Extracellular Donnan effectors are invariant in SM-CD, but at very small pump reserve (the danger zone), their concentration would affect steady state (Mehta et al., 2008). Because Coles et al. (2019) report protease release of extracellular matrix proteins in exercised mdx fibers, extracellular Donnan effector influences (particularly in connection with ischemia-related ΔpH; Hagberg, 1985) could be worth revisiting.

Nonosmotic Na+ loading, even in the danger zone

Fig. 9 B shows SM-CD as robust above ∼40% pump strength (pump reserve there: 4.4-fold) and deteriorating steeply below ∼20%, the danger zone where ATP consumption (=hyperpolarizing INaKpump) falls sharply as [Na+]i rises steeply. Vm-related nonlinearities create a vicious positive feedback via two novel flux-mechanism features (i.e., features negligible or not applicable in the 100% to ∼40% pump-strength range): (1) Na+ saturation of [Na+]i-sensitive hyperpolarizing Na+ extrusion (see Materials and methods; Fig. 4 A; see approach to ∼90 mM [Na+]i), and (2) undesirable recruitment of Nav channels to operational PNa. [Na+]i rises steeply as reduced hyperpolarizing INapKump fails to counteract the Nav-augmented depolarizing INaleak. These altered steady-state features pull SM-CD ever closer to defeat: i.e., spontaneous firing and cytotoxic swelling at X.

Nevertheless, right through the danger zone (until X), Na+ loading and K+ depletion remain well-matched. Thus, Cl influx and thus H2O influx stay small (note the y axis values for steady-state [Na+]i, [K+]i, and [Cl]i). Not shown, from 100%→X, impermeant cytoplasmic anions dilute slightly ([A]i: 149.6 mM→143.1 mM). Even at the saddle node (X), where [Na+]i = 88 mM, steady-state [Na+ + Cl] loading is so minor that osmo-balancing H2O entry has swollen SM-CD by only ∼4.5% (2,000 µm3→ ∼2,090 µm3).

Thus, [big PCl]-endowed SM-CD can sustain enormous low pump-strength–induced Na+ overloads with inconsequential water uptake. Weber et al. (2012), following the first use of 23Na-MRI with DMD patients, stressed the chronic nature of the overload, but could not assess if it signified cytotoxic swelling. More recently, using 23Na-proton-MRI, Gerhalter et al. (2019) showed that DMD patients’ chronic Na+ overload can occur in the absence of water T2 alterations, consistent with SM-CD modeling here.

Membrane damage: Too much Na+ leak in DMD

For SMFs, PK >> PNa, specifically (as modeled in SM-CD and LA-CD), PK:PNa is 1:0.03. If, in SMFs, nonselective cation channels with PK:PNa = 1:1 were to activate to an extent that doubled resting PNa (0.3 µm3/s→0.6 µm3/s; a 100% increase), total PK would increase just 3%, making this cation channel leak principally a Na+ leak. Both the hormonally regulated NALCN of various excitable cells (Lu et al., 2007; Cochet-Bissuel et al., 2014; Lutas et al., 2016; Philippart and Khaliq, 2018; Reinl et al., 2018) and the AChR channels of SMFs are nonselective cation channels serving as physiological Na+ leaks. To date, no SMF isoform of NALCN has been detected, but SMFs express various cation channels (Metzger et al., 2020), any of which, if overactive, would augment operational PNa. Unidentified overactive mechanosensitive cation channels in mdx fibers are a proposed therapeutic target for GsMtx4-based peptides (Franco and Lansman, 1990; Gnanasambandam et al., 2017; Ward et al., 2018). Like inappropriately active mdx AChR-cation channels (Carlson and Officer 1996), they are considered dangerous as Ca2+-entry paths (Yeung et al., 2005; Lansman, 2015; Ward et al., 2018), but if so, unavoidably, they would also be pathological Na+-leak channels (McBride et al., 2000; Yeung et al., 2003).

AChRs are identified, abundant, reportedly leaky in mdx (Carlson and Officer 1996), mechanosensitive and inhibitable by GsMtx4 (Pan et al., 2012), and known to exhibit spontaneous activity (Jackson et al., 1990), whose frequency could increase in damaged junctional mdx sarcolemma (Barrantes et al., 2010; Baenziger et al., 2017; Pratt et al., 2015; Kravtsova et al., 2020). We model leaky cation channels, therefore, using the SMF AChR channel PK:PNa ratio (1.11:1; Hille, 2001) but refer simply to leaky cation channels.

SMFs’ largest physiological Na+ influxes are via Nav1.4 channels bound via syntrophin to dystrophin (Gee et al., 1998; Fu et al., 2011). In mdx, sarcolemmal Nav1.4 density is subnormal, but gating appears normal (Mathes et al., 1991; Ribaux et al., 2001; Hirn et al., 2008). Hirn et al. (2008) show that in the 3 d after mechanically traumatic fiber isolation, 3 nM tetrodotoxin protects mdx fibers from Na+ loading and die off. In the bleb-damaged sarcolemma of DMD fibers (Fig. 1 B), Nav1.4 channels could exhibit left-shifted window conductance (Nav-CLS;Boucher et al., 2012; see Materials and methods; Fig. 4 F).

The next sections address these two classes of sarcolemma damage–mediated Na+ leaks: first Nav-CLS, then overactive cation channels.

Nav-CLS: Treacherous but not a sole cause of chronic Na+ overload

For CN-CD and for SM-CD subjected to increasing Nav-CLS (at pump strength = 100%), Fig. 10 depicts system excitability (normal, hypersensitive, spontaneous; red line, top) and the systems’ steady-state P-L/D values. The term affected channels 0.3 (AC = 0.3) means that damage-induced Nav-CLS affects 3/10th of the Nav channel bearing membrane (see Materials and methods); as per Wang et al. (2009), more intense bleb-type damage→↑CLS (in mV).

In CN-CD, Vrest (−65.5 mV) is close to firing threshold. Hypersensitivity (sloped line, top) is the only marked Nav-CLS pathology in CN-CD until, at Nav-CLS[0.3] = 9 mV, the shifted window conductance triggers spontaneous firing (not shown). By Nav-CLS[0.3] = 10 mV, spontaneous AP Na+ influx is overwhelming; CN-CD is cytotoxically swollen (off scale for the plots).

Though SM-CD Nav density is 3× higher, its hyperpolarized Vrest (−86 mV) protects against Nav-CLS[0.3]. SM-CD too exhibits ever-increasing hypersensitivity (top) as Nav-CLS intensity increases, but at, say, 10 mV, the Nav-CLS[0.3] has no other impact. At 20 mV, Vrest is depolarized almost imperceptibly. Beyond 23 mV, this steepens sharply. See Fig. 10 legend for details.

The SM-CD Nav-CLS damage range between zero Na+ loading and the onset of highly problematic sustained spontaneous firing (see [Na+]i panel, Fig. 10 B, i) is exquisitely narrow. Thus, Nav-CLS alone (i.e., in a full pump-strength system) could not explain chronic Na+ overload in DMD fibers.

Damage-induced hypersensitivity of Fig. 10 B, i (top), would promote unwanted spontaneous APs. This could be why the single-fiber electromyographic recordings of advanced-stage DMD patients (whose fibers would certainly not be at full pump strength) show “bizarre repetitive discharge bursts” (Trontelj and Stålberg, 1983; see also Yu et al., 2012). Nav-CLS would foster, too, the in situ erratic spontaneous firing and contractility reported for DMD patients (Ishpekova et al., 1999; Emeryk-Szajewska and Kopeć, 2008; Nojszewska et al., 2017); electromyography of mdx muscle shows abnormal spontaneous potentials and complex repetitive discharges like those of boys with DMD (Carter et al., 1992; Han et al., 2006).

Depolarization and Na+ loading due to leaky cation channels

Stretch injury to healthy (McBride et al., 2000) and mdx fibers causes sustained depolarization (Baumann et al., 2020). Channels rendered hypermechanosensitive by membrane damage (e.g., Wan et al., 1999) might intermittently activate in mdx fibers (Lansman 2015), or normally quiescent cation channels might activate chronically in damaged sarcolemma. Fig. 11 shows what damage-induced chronic activation of sarcolemmal cation channels would contribute to such situations. In SM-CD and LA-CD (and counterfactual WD-CD), normally quiescent cation channels are activated. Systems converge on their membrane-damaged steady states by ∼20 min off trajectories (after 20 min) mimic step-application of a cation channel–specific inhibitor.

Leak amplitude for the PK:PNa = 1.11:1 cation channel is established by making its PNa = 0.3 µm3/s. For SM-CD and LA-CD, this doubles operational PNa. WD-CD’s larger resting PK and PNa values diminish the cation leak’s relative impact. ATP consumption rises almost identically in each system, but for already high-cost Pump-Leak dominated WD-CD, the rise is ∼22%, while for Donnan dominated SM-CD and LA-CD, this cation leak nearly doubles the cost of steady state.

SM-CD depolarizes by 10 mV (Vrest, green square); pathophysiologically, 10 mV is consequential (↑system excitability), but the attendant <1 mM Na+ loading (green asterisk) would hardly register. Low–pump-strength LA-CD depolarizes by 12 mV, and its [Na+]i (∼12 mM) is measurably >3.7 mM (the healthy control value for a [LA-CD/leaky] versus [SM-CD/no leak] comparison). The elevated [LA-CD/leaky] Na+ influx is countered by almost-equal K+ loss; accordingly, net [Na+ + Cl + H2O] uptake (= osmotic Na+ loading) would give a mere 2% cellular edema (above the [SM-CD/no leak] control). A [LA-CD/leaky] scenario could, therefore, explain depolarized Na+-overloaded mdx fibers (see Table 1, items 2, 3, 4, 7, and 8) and would be consistent with DMD patients’ chronic nonosmotic [Na+]i overload.

Its elevated ATP consumption (arrow; 90 amol/s) notwithstanding, depolarized [SM-CD/leaky] still has a robust 6.6-fold pump reserve (= 566 amol/s / 90 amol/s). [LA-CD/leaky], however, has only a 1.9-fold pump reserve (= 30% × 566 amol/s/90 amol/s). While undesirable, 1.9 puts [LA-CD/leaky] in league (regarding vulnerability and robustness) with healthy CNs (pump reserve = 2.1-fold for CN-CD). In advanced DMD, where functional ischemia could initiate ischemic rundowns, leaky cation channels would speed rundown (↑INaleak) and compromise recovery (↑INaleak→↓pump reserve).

In summary, in a 100% pump-strength system, leaky cation channels, though strongly depolarizing, augment [Na+]i too little to be a plausible standalone explanation for DMD fibers’ chronic Na+ overload. In conjunction with low SMF pump strength, however, they would contribute to chronic nonosmotic Na+ overloads and amplify the loss of ion homeostatic robustness.

Low pump strength + repetitive stimulation + two damage-related Na+ leaks

Like rat soleus fibers, SM-CD briskly returns to steady state after firing 1,200 APs (Fig. 5); Fig. 12 A shows LA-CD handling this same stress test, albeit with a slower recovery. [LA-CD/leaky] (as per Fig. 11) also manages the stress test (not shown), as does [LA-CD/(Nav-CLS[0.3] = 10 mV)] (not shown). But, as Fig. 12 B, i, shows, the multiply damaged (i.e., [LA-CD/leaky/(Nav-CLS[0.3] = 10 mV)]) stress-tested Donnan dominated P-L/D system is overwhelmed. Having stabilized at a depolarized Vrest (−74 mV), it seems to handle the 1,200 APs, but at 10 s, when stimulation stops (expanded, Fig. 12 B, ii), the Vm(t) trajectory reveals that all is not well. It does not return to steady state. For ∼60 s, APs (∼20 Hz) of diminishing amplitude fire spontaneously: then the system converges on a pathological depolarized steady state that (not shown) is swollen and gradient depleted. From the limited perspective of Vm(t), it is unclear when, during the stress test, a lethal threshold was crossed. In this lethal bistability scenario, to know when threshold crossing occurred would require simultaneous monitoring of (A) INaKpump and (B) total depolarizing INaleak. Lethal threshold crossing occurs when B > A.

Na+ overload and DMD ion homeostatic jeopardy

23Na-proton-MRI (Gerhalter et al., 2017, 2019) will increasingly be used to noninvasively monitor DMD patients. Representing worsening DMD severity, Fig. 13 A thus plots SM-CD Na+ loads corresponding to diminishing pump strength and accumulating (damage-induced) Nav-CLS. The y axis indicates the lowest pump strength able to prevent spontaneous firing at Nav-CLS[0.3] intensity on the x axis. Fig. 13 B extracts values from Fig. 13 A to graphically emphasize tolerable and intolerable [Na+]i zones. Coordinates corresponding to a quiescent Vm (no spontaneous firing, so no contracture and no cytotoxic swelling) are deemed safe. Beyond that, coordinates are deemed lethal. The Volcell plot (inset in Fig. 13 A) shows safe zone swelling to be negligible. Fig. 13 C is a sketched (no further computations) reminder that leaky cation channels will diminish the safe [Na+]i zone, pulling it nearer the origin (green dot).

An overall message: for damaged SMFs, there is no one canonical safe [Na+]i-load value. As per Fig. 13 A, for SM-CD pump strengths approaching the saddle-node value (Fig. 9), high [Na+]i can be sustained, but only if there is no damage–Na+ leak. At the other extreme (near 100% pump strength), profound membrane-damage Na+ leak is tolerated with almost no Na+ loading, but any pump-strength reduction would bring on lethal spontaneous APs. Between these extremes, chronic [Na+]i overloads beneath the descending Na+ line, though undesirable, are tenable. Quiescent fibers close to the safe boundary would have little resilience: transient physiological demands on the P-L/D system (AP trains, use of secondary transporters, a bout of functional ischemia) could push them into the lethal zone.

DMD-patient muscle [Na+]i values assessed by 23Na-MRI would combine individual fibers with coordinates throughout the safe zone. Safe-zone fibers well to the right would be mildly [Na+]i loaded but hyperexcitable and thus vulnerable to unwanted (and lethal) excitation–contraction coupling (Claflin and Brooks, 2008). At left (Fig. 13 B), in heavily [Na+]i-loaded safe-zone fibers, vulnerability to Ca2+ necrosis via reverse operation of Na+/Ca2+ exchangers (as per Burr et al., 2014) would be an ever-present danger. Ca2+ necrosis and ion homeostatic failure are not mutually exclusive explanations for DMD fiber demise.

To address the impaired ion homeostasis of Duchenne MD, a generic ion homeostasis model for excitable SMFs, SM-CD, was devised, then tested under normal and DMD-like conditions. For SMFs, myonuclear domain size is the average cyto-volume transcriptionally served by one nucleus in the syncytium. SM-CD does not represent the syncytial SMF; it is one ion homeostatic unit that, disposed as many contiguous slices, would represent a syncytial SMF. Back-calculating from mice (e.g., Mantilla et al., 2008), SM-CDs’ 2,000 µm3 would roughly correspond to one myonuclear domain volume. Cell biologically, this is reasonable because SM-CDs’ 2,000 µm3 matches CN-CDs’ one neuronuclear domain.

SM-CD and CN-CD are excitable P-L/D systems. SMFs’ and neurons’ distinctive ion homeostatic P-L/D strategies have been evolutionarily tuned to their distinctive electrophysiological lifestyles: mostly quiescent, low excitability, and resilient during severe ion homeostatic emergencies (syncytial SMFs) and electrically agile, electrically versatile, highly excitable (neurons). SMFs’ low-impedance Donnan-dominated steady state is robust and inexpensive. Neurons’ high impedance Pump-Leak–dominated steady state is vulnerable and expensive. In ancestral (like modern) vertebrates, SM occupied the largest fractional tissue mass (Mink et al., 1981; Rolfe and Brown, 1997; Helfman et al., 2009; Johnston et al., 2011; Casane and Laurenti, 2013; Dutel et al., 2019; see Table S1); a low-cost steady state for this massive tissue is self-evidently advantageous. Neurons’ executive functions are indispensable for the whole organism; that, plus their small fractional body mass (<0.2% in ancestral vertebrates, ∼2% in humans), has, self-evidently, rendered their precariously costly ion homeostasis acceptable.

DMD ion homeostasis is simulated here as reduced pump strength and/or inappropriately active Na+-permeant ion channels. Ca2+ necrosis, considered the usual proximate cause for fiber loss in DMD, is indirectly addressed. Beyond DMD, pathological situations modeled here are relevant to healthy SMFs that experience traumatic/exercise injury, including situations arising as compartment syndrome.

Prospects

DMD molecular-therapy preclinical and clinical trials are in progress (Verhaart and Aartsma-Rus, 2019; Meng et al., 2020; Datta and Ghosh, 2020; Chemello et al., 2020; Wagner et al., 2021; Duan et al., 2021), and noninvasive monitoring of aspects of muscle ion homeostasis is increasingly feasible (e.g., Dahlmann et al., 2016; Mankodi et al., 2017; Forbes et al., 2020; Pennati et al., 2020; Zhang et al., 2020; Dietz et al., 2020; Sherlock et al., 2021). The generic theoretical framework outlined here for excitable SMF ion homeostasis should prove helpful in those contexts. DMD fibers’ operational pump strength (as modeled here) depends on the diverse factors mentioned in Fig. 1 C, some of which are therapeutic targets. As pump strength diminished and as fibers accumulated sarcolemma damage in DMD, our analysis indicates that Nav1.4 channels and leaky (Na+-permeant)–cation channels would be therapeutic targets.

Generic SMF ion homeostatic theory as captured by SM-CD (as is) should help frame discussion of (for example) fiber-type–specific single-fiber “snap-freeze” proteomics (Deshmukh et al., 2021), studies of fiber-type– and exertion-regimen–specific ClC-1 immunocytochemistry (Thomassen et al., 2018), and so on. For quantitative use in connection with data obtained, for example, by simultaneous monitoring of multiple SMF ions (Heiny et al., 2019; DiFranco et al., 2019), SM-CD needs to be up-graded (e.g., SMF appropriate V-gated channel kinetics and SMF-specific pumps; for the DMD context, this would extend to the different kinetic to diaphragm and postural muscle pump isoforms; Kravtsova et al., 2020; also see supplemental text).

Fig. 14, left, summarizes the “as-is machinery” of SM-CD, and at right, suggests what a next-stage generic SMF model could incorporate. The following section itemizes what as-is SM-CD reveals about how SMF ion homeostasis copes as well as it does under DMD conditions, and also how it would eventually fail.

DMD-related SMF ion homeostasis as seen via the [big PCl][small INaleak] model, SM-CD

Chronic low–pump strength

Chronic low–pump strength is tolerated because [small INaleak] → a small requirement for ATP-generated INaKpump at steady state. Thus, even at chronically low pump strengths (say, 30% LA-CD; Table 2), enough pump reserve can remain to handle moderate ionic perturbations (Fig. 7 C, iii; Fig. 11; and Fig. 12 A).

DMD functional ischemia

Transiently, DMD functional ischemia could take already low pump-strength fibers deep into the danger zone (Fig. 9 C). Longer term, the attendant membrane damage could ↑INaleak and ↓pump strength. A bout of functional ischemia to below a fiber’s saddle node would initiate ischemic rundown; the weaker the pump strength and the larger the INaleak, the faster the rundown (Fig. 6 C, inset plot). However, if that bout ended before spontaneous firing commenced (Fig. 7 C, iii), even pump strength–boosting factors as minor as massage for circulatory improvement (Carroll et al., 2020) will contribute measurably to DMD fiber survival (Fig. 9 C).

Even at ultra-low pump strength, Na+ loading is mostly nonosmotic

Fig. 9 shows how, even at the saddle node (X), with [Na+] loading massive, with [small INaleak] there contaminated by Nav window current, operational INaleak has remained small enough that Volcell has increased only a few percent. Hammon et al. (2015) show that during anaerobic exercise, healthy SMFs exhibit a water-independent (i.e., nonosmotic) ↑[Na+]i; this seems broadly congruent with anoxic and rundown/recovery behavior for SM-CD and LA-CD (Fig. 7).

As per 23Na-proton-MRI, chronic SMF Na+ overload is nonosmotic

In SM-CD, PCl >> PK >>> PNa with Vrest = ECl. INaleak is so small that during stress testing and rundowns, electro-osmo-balance is maintained with very little [Na+ + Cl + H2O] entry. In such situations, if spontaneous firing began (Fig. 6; and Fig. 7 A, i), the concurrence, for actual fibers, of contracture with osmotic Na+ loading would presumably destroy them. Such fibers would therefore not contribute to whole-muscle 23Na-proton-MRI signals (see Fig. 13). If SMFs’ V-gated channels were fully inactivated, osmotic Na+ loading could develop safely (see, e.g., Fig. 8 [SM*-CD] at ∼50 min: with ENa ∼0 mV (maximal Na+ loading), swelling is only (2,400/2,000) 1.2-fold (for reference, after exhaustive exercise, 1.2-fold is the upper limit for fiber swelling [Sjøgaard et al., 1985, recounted by Fraser and Huang, 2004]). H2O loading lags Na+ loading there, [big PCl] and very big [PH2O] notwithstanding. Why? Because SM*-CD is a minimal P-L/D system, so ECl tracks Vm (keeping the driving force on Cl small; see ICl, Fig. 8). Unlike nonosmotic chronic Na+ overload in DMD, overactive Na-K-2Cl co-transport would produce osmotic Na+ loading (Fig. 14 E).

Chronic depolarization without swelling

At very low pump strengths, there is insufficient hyperpolarizing INaKpump to maintain normal Vrest (e.g., see Vm for 20% pump strength; Fig. 9 B), so (in the danger zone below ∼20% pump strength) depolarization →↑chronic Nav subthreshold activity. Danger zone depolarization and reduced ATP consumption therefore coincide, making long-term fiber viability in the danger zone improbable. More likely, chronically depolarized DMD fibers have leaky cation channels plus more mildly reduced pump strength, like LA-CD in Fig. 11, whose depolarized state recruits some subthreshold Nav activity along with the leaky cation channels. Such fibers would be Na+ loaded but barely swollen. If, additionally, there was Nav-CLS damage (Figs. 10 and 13), higher pump strengths would be required for a fiber to remain safe.

Erratic spontaneous contractility

Erratic spontaneous contractility is explicable if damaged DMD sarcolemma was hypersensitized by Nav-CLS (Fig. 10 B, i, top) and if fibers were chronically depolarized as per Chronic depolarization without swelling. Transient cation channel leaks could then trigger spontaneous AP bursts (i.e., not attributed to end plate input). Though unwanted contractile events are undesirable, brief episodes as just described would be manageable, ion homeostatically, for chronically low pump-strength fibers still tolerably far from a saddle node (X).

Low tolerance for sustained exertion

During exertion, fiber ATP is needed for both contractile and ion homeostatic processes. DMD’s functional ischemia occurs for lack of the healthy fiber vasodilation that keeps operational pump strength safely high during exertion (Fig. 9 C).

SMF circadian rhythms could imperil already-compromised DMD respiratory muscles

DMD patients minimize daytime muscle exertion, but nights too are problematic. During rapid eye movement sleep, respiratory profiles are acutely compromised (Nozoe et al., 2015; Hartman et al., 2020; MacKintosh et al., 2020). Muscle tissue sleep rhythms diminish SMF glucose availability and oxidative capacity (Harfmann et al., 2015; van Moorsel et al., 2016; Ehlen et al., 2017); the reduced pump strength could take DMD diaphragm and thoracic fibers to their lethal saddle nodes (Fig. 9 C).

Safe Na+ overload: What can be deemed safe varies with the mix of deficits

Due to DMD membrane damage, Nav channels may have Nav-CLS of unknown intensity. Thus, no one Na+-overload level can reliably be designated safe (Fig. 13). The paucity of human SMF data (plus its disparity with respect to mouse data; Table 1) make it hard to assess the [Na+]i values of Fig. 13, as does the fact that SM-CD pump kinetics are not those of human (or mouse) SMF pumps (Fig. 14). Anecdotally, Dahlmann et al. (2016) used 23Na-MRI to monitor a young athlete’s muscle injury and recovery (control – the noninjured contralateral); this showed at 0, 2, and 8 wk ([Na+]i in mM), control/injured, 18/44, 19/38, and 19/22, respectively. By comparison (as per Table 1), control/DMD readings are ([Na+]i in mM) 25/38 (Weber et al., 2011) and 16/26 (Gerhalter et al., 2019). The global point of Fig. 13 is that nonlinear interplays among pump strength, Vrest, Nav window current, and damage-induced ↑PNa (from Nav-CLS and leaky cation channels) diminish safe Na+-overload levels nonlinearly. The down-sloping Na+ line (Fig. 13, B and C) implies that while, say, 80 mM would be safe if the sole DMD deficit is low pump strength, 30 mM could be lethal if there was also, say, ∼20 mV of Nav-CLS[0.3]. If, during tissue remodeling of damaged depolarized SMFs, Nav channels were to chronically slow inactivate (Webb et al., 2009), safety would increase (Fig. 8).

Ca2+ necrosis: An elusive consequence of failed ion homeostasis

Here, for low pump-strength systems, Ca2+ necrosis can be taken as implied by the onset of spontaneous firing (→Ca2+-mediated excitation–contraction coupling; in ATP-depleted fibers, contraction would be irreversible). Rundowns (Figs. 6 and 7) elicit such firing. Physiological ENa depletion from AP trains (Fig. 12 B) or, say, an intense bout of cotransporter activity (see Fig. 14 E) could take DMD fibers already coping with insult/injury to a saddle-node threshold (thence to Ca2+ necrosis). Conditions for fiber demise are thus diverse. Moreover, though the criterion for the onset of lethal spontaneous firing is simple, hyperpolarizing INaKpump < depolarizing INaleak(total), knowing that a threshold has been crossed can be elusive. Even computationally, with multiple nonlinear factors affecting Na+ fluxes (as per Safe Na+ overload), recognizing threshold crossing is tricky in the time domain. Depending on the initial conditions before injured SM-CD encountering an unstable threshold (in parameter space), irreversible crossing could happen within milliseconds or might take hours. Consider Fig. 6 (a tourniquet, say, applied at t = 0 to a healthy fiber): this system is now destabilized, but an observer monitoring Vm(t) between, say, 300–320 min would likely report a seriously depolarized but stable Vrest. Consider Fig. 12 B: during the stress test, though Vm(t) seems unproblematic, during those 1,200 Aps, the system destabilizes. Recordings that report on SMF [Ca2+]i(t) are typically visually striking, and they are more easily accomplished recordings of [Na+]i(t) or even Vm(t). Instances of fiber loss reported as Ca2+ necrosis could therefore reflect ion homeostatic failure. Advances in cell physiological instrumentation could enhance P-L/D investigations of SMFs in the DMD context: it is now possible, with a four-electrode method (Heiny et al., 2019), to follow multi-ion dynamics and transport processes under voltage or current clamp, even monitoring [Cl]i accurately (DiFranco et al., 2019).

Therapeutics and optimizing the P-L/D process in DMD SMFs

Diminishing DMD fiber loss by bolstering SMFs’ ion homeostasis robustness (i.e., moving as close as possible to the green dots of Figs. 9, 10, and 13) would require (1) optimizing operational pump strength by countering the factors listed in Fig. 1 C, and (2) minimizing Na+ leaks. Examples for (1) are up-regulating pumps and/or modifying their kinetics (Breitenbach et al., 2016; Glemser et al., 2017; Raman et al., 2017), ↑vascular sufficiency (e.g., angiogenesis; Verma et al., 2019; Podkalicka et al., 2019), dietary supplementation that increases vascular density (Banfi et al., 2018), ↑vasodilation (e.g., NO reagents; Rebolledo et al., 2016; Patel et al., 2018), and massage (Saxena et al., 2013; Carroll et al., 2020). Insofar as stressors (e.g., during ischemic preconditioning; Rongen et al., 2002) cause myofibers to preemptively up-regulate their pump strength (↑pump reserve), encouraging this (Glemser et al., 2017) could help prevent threshold catastrophes. Examples for (2) are suppression of the danger zone’s treacherous subthreshold Nav1.4 current, e.g., via ranolazine and/or anticonvulsant agents that enhance Nav1.4 slow inactivation in SMFs (Lorusso et al., 2019; Skov et al., 2017), neither of which is (yet) in use for DMD. Long-available pharmacological tools could establish whether leaky AChR channels (Carlson and Officer 1996; Pan et al., 2012) contribute to DMD fiber depolarization/Na+ loading.

Sarcolemmal damage causes Na+ loading via Na+-permeant (tetrodotoxin-insensitive) channels (Yeung et al., 2003) thought to be the overactive (and adventitiously mechanosensitive) cation channels of mdx fibers (Yeung et al., 2005; Ward et al., 2018). The scenario resembles LA-CD/leaky (Fig. 11). In healthy SMFs, does that cation channel do for SMFs what NALCNs do for neurons and smooth muscle (Lutas et al., 2016; Philippart and Khaliq, 2018)? In other words, is it SMFs’ physiological PNa? Perhaps the membrane-active compounds coming into use for DMD patients (Houang et al., 2018; Sreetama et al., 2018; Conklin et al., 2018) will stabilize a leaky SMF PNa, but with SMF PNa unidentified, this would be difficult to test. SMF’s small-valued PNa has been overlooked but, as emphasized here, its smallness makes it powerful, not trivial. That smallness is an energetically pivotal feature of the human body’s largest tissue. Like the process underway for its neuronal counterpart (NALCN, e.g., Kang et al., 2020) molecular identification and structural characterization of SMF PNa would open routes to rational development of inhibitors and modulators.

Jeanne M. Nerbonne served as editor.

We acknowledge financial support from Natural Sciences and Engineering Council (Canada; grant RGPIN-06835-2018 to B. Joos) and support from the Ottawa Hospital Research Institute (to C.E. Morris).

The authors declare no competing financial interests.

Author contributions: C.E. Morris: conceptualization, model validation, figure preparation, and manuscript writing and revising; B Joos: conceptualization, model development, coding, optimizing, and validating the charge difference approach computational models, implementing the models and curating the outputs, writing Materials and methods, and manuscript editing; and J.J. Wheeler: contributing to coding and data presentation.

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